a. $A=(2x-3)^2-(x+1)(x+5)+2$
$A=(4x^2-12x+9)-(x^2+6x+5)+2$
$A=4x^2-12x+9-x^2-6x-5+2$
$A=3x^2-18x+6$
$A=3(x^2-6x+2)$
$A=3(x^2-6x+9-7)$
$A=3[(x-3)^2-7]$
$A=3(x-3)^2-21$
Vì $3(x-3)^2 \ge 0\; \forall x$
$\Rightarrow 3(x-3)^2-21 \ge -21\; \forall x$
Vậy $\min A=-21$ khi $x-3 = 0 \Leftrightarrow x=3$
b. $P=x^2-2xy+6y^2-12x+2y+45$
$P=\left(x^2-2xy+y^2-12x+12y+36\right)+\left(5y^2-10y+5\right)+4$
$P=\left[\left(x-y\right)^2-12\left(x+y\right)+6^2\right]+5\left(y^2-2y+1\right)+4$
$P=\left(x-y+6\right)^2+5\left(y-1\right)^2+4$
Vì$ \left(x-y+6\right)^2+5\left(y-1\right)^2\ge 0 \; \forall x$
$\Rightarrow \left(x-y+6\right)^2+5\left(y-1\right)^2+4 \ge 4 \; \forall x$
Vậy $\min P=4$ khi $\begin{cases}x-y+6=0\\y-1=0\end{cases} \Leftrightarrow \begin{cases}x=7\\y=1\end{cases}$
